We start by setting up the model-categorical background used throughout this paper.
Partially following [1]} and [2]}, we consider the category of simplicial presheaves on cartesian spaces with its canonical projective and injective model structures.
In analogy with \({\mathbb {A}}^1\) -local homotopy theory, we localise this category at all the morphisms \(c \times \mathbb {R}\rightarrow c\) , where \(c\)  is any cartesian space and where the morphism collapses the real line.
Extending ideas from [2]}, we establish several Quillen equivalences of this localised model category with the categories of simplicial sets and topological spaces.
